Recently Ha et al. [1]} introduced the local as well
as global smoothness indicators based on \(L_{1}\) -norm approach and
termed this scheme as WENO-NS. The WENO-NS scheme provides an improved
behavior compared to other fifth-order WENO schemes for the problems
which contain discontinuities. It is well known that the smoothness
indicators constructed based on \(L_{1}-\) norm approach may lead to
provide loss of regularity of the solution. To overcome this difficulty,
the authors constructed local smoothness indicators by developing
an approximation method to derivatives with higher accuracy and introduce
a parameter \(\xi \)  in calculating the local smoothness indicators
to the tradeoff between the accuracy around the smooth and discontinuous
regions. The global smoothness indicator is constructed as an average
of the smoothness of a global stencil data and a middle stencil information
through a mapping function which satisfies desirable sufficient condition
so that the scheme achieves required order of accuracy.
